Conjugacy classes of symplectic group $\mathrm{Sp}(4,q)$

Keep in mind that it's tricky to assemble a character table for all finite groups of Lie type in a family, and in particular the thesis work by Srinivasan at Manchester (supervised by J.A. Green) excluded the prime $p=2$ (later filled in by H. Enomoto here. Moreover, tables often have errors; here there are small errors found by A. Pryzgocki here. And as the comment by Bullet51 shows, not everything is straightforward: at the time there was mainly a paper by G.E. Wall on the conjugacy classes of finire classical groups. Her thesis work was influential in the later development of sophisticated methods by Deligne and Lusztig (1976). But it is still difficult to summarize character tables for an entire Lie family.

Concerning your attempt to work out the classes in a concrete way, I'm not sure exactly what goes wrong. But this does illustrate the perils of trying to be too computational: such an approach will be troublesome for larger matrices.